Some Optimal Classes of Estimators Based on Multi-Auxiliary Information
<p>Graph of (<b>a</b>) Multiples of <math display="inline"><semantics> <mover accent="true"> <mi>Y</mi> <mo stretchy="false">¯</mo> </mover> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>M</mi> <mi>S</mi> <mi>E</mi> </mrow> </semantics></math> of estimator <math display="inline"><semantics> <msub> <mi>T</mi> <mrow> <mi>b</mi> <msub> <mi>k</mi> <mn>1</mn> </msub> </mrow> </msub> </semantics></math> (<b>b</b>) Multiples of <math display="inline"><semantics> <msub> <mo>Φ</mo> <mn>1</mn> </msub> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>M</mi> <mi>S</mi> <mi>E</mi> </mrow> </semantics></math> of estimator <math display="inline"><semantics> <msub> <mi>T</mi> <mrow> <mi>b</mi> <msub> <mi>k</mi> <mn>1</mn> </msub> </mrow> </msub> </semantics></math> for the Population 1.</p> "> Figure 2
<p>Graph of (<b>a</b>) Multiples of <math display="inline"><semantics> <mover accent="true"> <mi>Y</mi> <mo stretchy="false">¯</mo> </mover> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>M</mi> <mi>S</mi> <mi>E</mi> </mrow> </semantics></math> of estimator <math display="inline"><semantics> <msub> <mi>T</mi> <mrow> <mi>b</mi> <msub> <mi>k</mi> <mn>2</mn> </msub> </mrow> </msub> </semantics></math> (<b>b</b>) Multiples of <math display="inline"><semantics> <mfrac> <msubsup> <mi>B</mi> <mn>2</mn> <mn>2</mn> </msubsup> <msub> <mi>A</mi> <mn>2</mn> </msub> </mfrac> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>M</mi> <mi>S</mi> <mi>E</mi> </mrow> </semantics></math> of estimator <math display="inline"><semantics> <msub> <mi>T</mi> <mrow> <mi>b</mi> <msub> <mi>k</mi> <mn>2</mn> </msub> </mrow> </msub> </semantics></math> for the Population 1.</p> "> Figure 3
<p>Graph of (<b>a</b>) Multiples of <math display="inline"><semantics> <mover accent="true"> <mi>Y</mi> <mo stretchy="false">¯</mo> </mover> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>M</mi> <mi>S</mi> <mi>E</mi> </mrow> </semantics></math> of estimator <math display="inline"><semantics> <msub> <mi>T</mi> <mrow> <mi>b</mi> <msub> <mi>k</mi> <mn>3</mn> </msub> </mrow> </msub> </semantics></math> (<b>b</b>) Multiples of <math display="inline"><semantics> <mfrac> <msubsup> <mi>B</mi> <mn>3</mn> <mn>2</mn> </msubsup> <msub> <mi>A</mi> <mn>3</mn> </msub> </mfrac> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>M</mi> <mi>S</mi> <mi>E</mi> </mrow> </semantics></math> of estimator <math display="inline"><semantics> <msub> <mi>T</mi> <mrow> <mi>b</mi> <msub> <mi>k</mi> <mn>3</mn> </msub> </mrow> </msub> </semantics></math> for the Population 1.</p> ">
Abstract
:1. Introduction
- (i).
- to provide some optimal classes of estimators based on multi-auxiliary information having the maximum relative efficiency and the least ,
- (ii).
- to provide a comparative study theoretically,
- (iii).
- to provide an extensive simulation study and a real data application for comparison with the existing estimators.
2. Methodology and Notations
3. Review
4. Proposed Estimators
4.1. Proposed Estimators Using BAI
4.2. Proposed Estimators Using MAI
4.3. Bias and Minimum MSE of the Proffered Estimators
5. Optimality Conditions
6. Computational Study
6.1. Numerical Study Utilizing Real Populations
6.2. Simulation Study Utilizing Real Populations
- (i).
- Take the natural populations reported in Section 6.1.
- (ii).
- Select a ranked set sample with r = 4 and m = 3 so that size units using RSS from every population.
- (iii).
- Obtain the necessary summary statistics.
- (iv).
- Repeat the aforementioned steps 15,000 times and tabulate the and PRE for each population.
6.3. Simulation Study Using Artificially Drawn Populations
- (i).
- Quantify simple random samples from the parent population with replacement.
- (ii).
- Extract ranked set sample by utilizing the RSS methodology.
- (iii).
- Calculate the required statistics.
- (iv).
- Calculate the and PRE of several estimators by repeating the prior points 15,000 times.
6.4. Discussion of Computational Findings
- (i).
- The numerical findings tabulated using real populations are displayed in Table 1 that show that the proffered estimators repress the existing estimators , , , , , , and by lowest and highest PRE.
- (ii).
- The simulation findings displayed in Table 3 for real populations show that the proffered estimators are found to be superior than the existing estimators.
- (iii).
- The simulation findings displayed in Table 4 for the artificially generated normal population exhibit that the proffered estimators attain the lowest and highest PRE regarding the existing estimators for various values of correlation coefficients. Moreover, it is also observed that the decreases as the correlation coefficient increases and vice versa in the sense of PRE.
- (iv).
- From Table 5 consisting of the findings of the simulation study for artificially generated asymmetric populations, the proffered estimators repress the conventional estimators with the lowest and highest PRE.
- (v).
- Moreover, from the findings of Table 1, Table 4 and Table 5 consisting of a numerical study using real populations and a simulation study using artificially generated populations, the proffered estimator is established to be superior among the proffered classes of estimators whereas the proffered estimator is found to be superior among the proffered classes of estimators from the results of Table 3 based on simulation study using real populations.
7. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A
Appendix B
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Estimators | Population 1 | Population 2 | Population 3 | Population 4 | ||||
---|---|---|---|---|---|---|---|---|
MSE | PRE | MSE | PRE | MSE | PRE | MSE | PRE | |
1133.892 | 100.000 | 0.000532 | 100.000 | 3,027,926 | 100.000 | 1,876,164 | 100.000 | |
1997.891 | 56.754 | 0.000652 | 81.633 | 3,863,125 | 78.380 | 5,765,568 | 32.540 | |
1082.504 | 104.747 | 0.000103 | 512.413 | 212,002 | 1428.252 | 228,686 | 820.410 | |
1082.504 | 104.747 | 0.000103 | 512.413 | 212,002 | 1428.252 | 228,686 | 820.410 | |
1078.606 | 105.125 | 0.000210 | 252.303 | 355,522 | 851.682 | 223,294 | 840.219 | |
2034.186 | 55.741 | 0.001860 | 28.609 | 11,959,036 | 25.319 | 4,111,339 | 45.633 | |
1997.496 | 56.765 | 0.000150 | 353.911 | 3,859,104 | 78.461 | 5,334,283 | 35.171 | |
1994.011 | 56.864 | 0.000439 | 121.090 | 3,834,538 | 78.964 | 1,412,015 | 132.871 | |
1996.164 | 56.803 | 0.000493 | 107.817 | 3,842,672 | 78.797 | 2,032,861 | 92.291 | |
3618.840 | 31.333 | 0.000572 | 92.942 | 27,886,266 | 10.858 | 15,976,676 | 11.743 | |
1238.466 | 91.556 | 0.000488 | 108.913 | 6,838,612 | 44.276 | 13,330,568 | 14.074 | |
1082.504 | 104.747 | 0.000103 | 512.413 | 212,002 | 1428.252 | 228,686 | 820.410 | |
1262.336 | 89.824 | 0.000585 | 90.885 | 17,524,939 | 17.277 | 57,190,391 | 3.280 | |
599.449 | 189.155 | 0.000103 | 513.702 | 209,820 | 1443.107 | 214,423 | 874.981 | |
555.289 | 204.198 | 0.000103 | 514.386 | 201,387 | 1503.533 | 104,759 | 1790.933 | |
586.707 | 193.263 | 0.000103 | 513.414 | 206,611 | 1465.516 | 149,486 | 1255.072 |
Conditions | Numerical Illustration Using Population 1 |
---|---|
Condition 6 | 0.5537 > 0.1559 |
Condition 7 | 0.5866 > 0.1559 |
Condition 7 | 0.5632 > 0.1559 |
Condition 8 | 0.5537 > −0.4872 |
Condition 9 | 0.5866 > −0.4872 |
Condition 9 | 0.5632 > −0.4872 |
Condition 10 | 0.5537 > 0.1941 |
Condition 11 | 0.5866 > 0.1941 |
Condition 11 | 0.5632 > 0.1941 |
Condition 12 | 0.5537 > 0.2666 |
Condition 13 | 0.5866 > 0.2666 |
Condition 13 | 0.5632 > 0.2666 |
Condition 14 | 0.5537 > −0.5142 |
Condition 15 | 0.5866 > −0.5142 |
Condition 15 | 0.5632 > −0.5142 |
Condition 16 | 0.5537 > −0.4869 |
Condition 17 | 0.5866 > −0.4869 |
Condition 17 | 0.5632 > −0.4869 |
Condition 18 | 0.5537 > −0.4843 |
Condition 19 | 0.5866 > −0.4843 |
Condition 19 | 0.5632 > −0.4843 |
Condition 20 | 0.5537 > −0.4859 |
Condition 21 | 0.5866 > −0.4859 |
Condition 21 | 0.5632 > −0.4859 |
Condition 22 | 0.5537 > −1.6939 |
Condition 23 | 0.5866 > −1.6939 |
Condition 23 | 0.5632 > −1.6939 |
Condition 24 | 0.5537 > 0.0780 |
Condition 25 | 0.5866 > 0.0780 |
Condition 25 | 0.5632 > 0.0780 |
Condition 26 | 0.5537 > −0.0136 |
Condition 27 | 0.5866 > −0.0136 |
Condition 27 | 0.5632 > −0.0136 |
Estimators | Population 1 | Population 2 | Population 3 | Population 4 | ||||
---|---|---|---|---|---|---|---|---|
MSE | PRE | MSE | PRE | MSE | PRE | MSE | PRE | |
1090.342 | 100.000 | 0.000492 | 100.000 | 2,860,741 | 100.000 | 17,71,020 | 100.000 | |
2239.336 | 48.690 | 0.001346 | 36.544 | 8,330,842 | 34.339 | 4,055,247 | 43.672 | |
891.551 | 122.297 | 0.000406 | 121.148 | 2,362,351 | 121.097 | 1,463,237 | 121.034 | |
891.551 | 122.297 | 0.000406 | 121.148 | 2,362,351 | 121.097 | 1,463,237 | 121.034 | |
907.034 | 120.209 | 0.000416 | 118.052 | 2,419,359 | 118.243 | 1,492,140 | 118.690 | |
1234.764 | 88.303 | 0.000640 | 76.830 | 3,718,263 | 76.937 | 1,959,909 | 90.362 | |
2238.959 | 48.698 | 0.000575 | 85.420 | 8,327,452 | 34.353 | 3,985,200 | 44.439 | |
2238.008 | 48.719 | 0.000496 | 99.112 | 8,316,689 | 34.397 | 3,783,187 | 46.812 | |
2238.632 | 48.705 | 0.000492 | 99.841 | 8,320,640 | 34.381 | 3,833,795 | 46.194 | |
2250.034 | 48.458 | 0.000492 | 99.825 | 8,222,207 | 34.792 | 3,822,069 | 46.336 | |
5070.206 | 21.504 | 0.000495 | 99.243 | 4,947,819 | 57.818 | 4,080,352 | 43.403 | |
891.551 | 122.297 | 0.000406 | 121.148 | 2,362,351 | 121.097 | 1,463,237 | 121.034 | |
4344.361 | 25.097 | 0.000482 | 101.874 | 4,504,028 | 63.515 | 3,793,459 | 46.686 | |
617.772 | 176.495 | 0.000401 | 122.552 | 2,103,387 | 136.006 | 1,181,170 | 149.937 | |
652.059 | 167.215 | 0.000401 | 122.393 | 2,143,901 | 133.436 | 1,217,474 | 145.466 | |
605.853 | 179.968 | 0.000401 | 122.580 | 2,095,924 | 136.490 | 1,169,544 | 151.428 |
0.5 | 0.7 | 0.9 | 0.5 | 0.7 | 0.9 | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
0.3 | 0.5 | 0.7 | 0.5 | 0.7 | 0.9 | |||||||
0.1 | 0.3 | 0.5 | 0.5 | 0.7 | 0.9 | |||||||
Estimators | MSE | PRE | MSE | PRE | MSE | PRE | MSE | PRE | MSE | PRE | MSE | PRE |
1.191 | 100.000 | 1.181 | 100.000 | 1.201 | 100.000 | 1.173 | 100.000 | 1.171 | 100.000 | 1.291 | 100.000 | |
1.874 | 63.547 | 1.429 | 82.628 | 1.000 | 120.038 | 1.700 | 68.992 | 1.262 | 92.810 | 0.819 | 157.650 | |
0.882 | 135.073 | 0.703 | 167.892 | 0.462 | 259.853 | 0.750 | 156.375 | 0.512 | 228.751 | 0.201 | 642.228 | |
0.882 | 135.073 | 0.703 | 167.892 | 0.462 | 259.853 | 0.750 | 156.375 | 0.512 | 228.751 | 0.201 | 642.228 | |
0.919 | 129.512 | 0.758 | 155.736 | 0.534 | 224.604 | 0.793 | 147.882 | 0.558 | 209.665 | 0.225 | 572.178 | |
1.983 | 60.069 | 2.506 | 47.129 | 3.026 | 39.685 | 2.536 | 46.265 | 3.006 | 38.973 | 3.642 | 35.447 | |
1.829 | 65.114 | 1.390 | 84.957 | 0.967 | 124.081 | 1.652 | 70.996 | 1.220 | 95.971 | 0.790 | 163.314 | |
1.621 | 73.465 | 1.214 | 97.245 | 0.827 | 145.212 | 1.432 | 81.915 | 1.034 | 113.321 | 0.666 | 193.752 | |
1.368 | 87.033 | 1.025 | 115.166 | 0.703 | 170.631 | 1.184 | 99.029 | 0.850 | 137.720 | 0.565 | 228.465 | |
2.491 | 47.815 | 2.896 | 40.780 | 3.333 | 36.030 | 2.823 | 41.563 | 3.224 | 36.339 | 3.827 | 33.737 | |
1.027 | 115.934 | 0.874 | 135.082 | 0.698 | 171.973 | 0.913 | 128.480 | 0.696 | 168.154 | 0.430 | 299.966 | |
0.882 | 135.073 | 0.703 | 167.892 | 0.462 | 259.853 | 0.750 | 156.375 | 0.512 | 228.751 | 0.201 | 642.228 | |
1.036 | 114.920 | 0.885 | 133.353 | 0.745 | 161.043 | 0.925 | 126.775 | 0.740 | 158.320 | 0.717 | 179.937 | |
0.871 | 136.753 | 0.696 | 169.588 | 0.459 | 261.612 | 0.742 | 157.996 | 0.508 | 230.384 | 0.200 | 643.787 | |
0.870 | 136.881 | 0.695 | 169.820 | 0.458 | 261.778 | 0.742 | 158.128 | 0.508 | 230.440 | 0.200 | 643.923 | |
0.870 | 136.787 | 0.696 | 169.591 | 0.459 | 261.521 | 0.742 | 158.020 | 0.508 | 230.380 | 0.200 | 643.849 |
Estimators | Population 2 | Population 3 | Population 4 | Population 5 | ||||
---|---|---|---|---|---|---|---|---|
MSE | PRE | MSE | PRE | MSE | PRE | MSE | PRE | |
0.151 | 100.000 | 0.164 | 100.000 | 0.050 | 100.000 | 1.154 | 100.000 | |
0.292 | 51.603 | 0.315 | 52.239 | 0.473 | 10.694 | 1.252 | 92.140 | |
0.085 | 176.481 | 0.079 | 206.521 | 0.026 | 189.187 | 0.452 | 255.040 | |
0.085 | 176.481 | 0.079 | 206.521 | 0.026 | 189.187 | 0.452 | 255.040 | |
0.091 | 164.875 | 0.086 | 189.816 | 0.028 | 175.540 | 0.510 | 226.182 | |
0.406 | 37.171 | 0.497 | 33.106 | 0.162 | 31.113 | 2.897 | 39.842 | |
0.212 | 71.074 | 0.213 | 77.121 | 0.045 | 110.216 | 1.137 | 101.466 | |
0.135 | 111.316 | 0.135 | 121.326 | 0.039 | 127.234 | 0.886 | 130.236 | |
0.123 | 122.502 | 0.126 | 130.135 | 0.076 | 66.501 | 0.827 | 139.559 | |
0.312 | 48.274 | 0.379 | 43.395 | 0.146 | 34.669 | 2.743 | 42.081 | |
0.126 | 118.973 | 0.132 | 124.488 | 0.101 | 50.056 | 0.806 | 143.069 | |
0.085 | 176.481 | 0.079 | 206.521 | 0.026 | 189.187 | 0.452 | 255.040 | |
0.143 | 105.453 | 0.152 | 107.739 | 0.119 | 42.234 | 0.878 | 131.473 | |
0.084 | 179.351 | 0.078 | 209.808 | 0.025 | 198.876 | 0.448 | 257.327 | |
0.083 | 180.143 | 0.077 | 211.013 | 0.024 | 204.165 | 0.448 | 257.471 | |
0.084 | 179.430 | 0.078 | 209.883 | 0.025 | 199.719 | 0.448 | 257.197 |
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Bhushan, S.; Kumar, A.; Alsadat, N.; Mustafa, M.S.; Alsolmi, M.M. Some Optimal Classes of Estimators Based on Multi-Auxiliary Information. Axioms 2023, 12, 515. https://doi.org/10.3390/axioms12060515
Bhushan S, Kumar A, Alsadat N, Mustafa MS, Alsolmi MM. Some Optimal Classes of Estimators Based on Multi-Auxiliary Information. Axioms. 2023; 12(6):515. https://doi.org/10.3390/axioms12060515
Chicago/Turabian StyleBhushan, Shashi, Anoop Kumar, Najwan Alsadat, Manahil SidAhmed Mustafa, and Meshayil M. Alsolmi. 2023. "Some Optimal Classes of Estimators Based on Multi-Auxiliary Information" Axioms 12, no. 6: 515. https://doi.org/10.3390/axioms12060515